Is it possible to animate a mesh's geometry without using Shape Keys?
AnimAll Addon

Yes, you can do this very easily by using the AnimAll addon. "AnimAll" stands for "Animate All" because it enables you to animate many things that cannot normally be animated in Blender. 

Here's how to use it:


First, open your User Preferences:
Info header → File → User Preferences  

 
Click on "Addons" and in the search box, type animall, then enable it by clicking on the Enable Addon checkbox on the far right hand side of the addon:
(NOTE: If you want to keep AnimAll enabled, click Save User Settings before closing the Preferences window)   


Now select the mesh you'd like to animate and go into Edit mode.
Now in the Tools panel (T), find the AnimAll section and enable Points -- this will allow you to animate the vertices of your mesh:
(NOTE: If you are using 2.70, AnimAll will be in the Animation tab in the Tools panel. If you are using 2.69, it will be down at the very bottom of the Tools panel) 


In the Timeline window, make sure the green frame cursor is set to the frame you want the animation to begin. Here, it's set to frame 1:  


Now in the AnimAll options, click the Insert (Keyframes) button.
(NOTE: This will insert keyframes for every single vertex in the mesh, regardless of whether they are selected or not, so if the mesh has lots of vertices, it may take a while before Blender is done inserting all the keyframes)  


In the Timeline window, move to the next important frame in your animation. Here, I'm moving to frame 75: 

 
Now alter your mesh's vertices as you like:  


Now insert keyframes for the vertices by again clicking Insert (Keyframes) button in AnimAll.
Now exit Edit mode and you can preview your animation by either clicking the Play button (  ) in the Timeline window or with ALTA:  






NOTE: The tutorial video on this page describes how to use all the many other powerful features of AnimAll.